The geometry of the space of SRTs can be used to define an average PDF of a sample of PDFs, which allows us to efficiently summarize and visualize the sample. The average PDF can be computed using a generalized version of a mean on a metric space called the Karcher mean [1]}, [2]}. Suppose we have \(n\)  pdfs \(f_1,\ldots ,f_n\)  and their corresponding SRTs \(h_1,\ldots ,h_n\) . The sample Karcher mean \(\bar{h}\in \mathcal {H}\)  is defined as the minimizer of the Karcher variance \(\rho (\bar{h}) \propto \sum \limits _{i=1}^n d(h_i,\bar{h})^2\) , i.e., \(\bar{h} = \text{argmin}_{h \in \mathcal {H}} ~\rho (h)\) . Algorithm REF  presents a gradient-based approach to compute the Karcher mean on \(\mathcal {H}\) . The Karcher mean of the sample PDFs is an intrinsic average that is computed directly on \(\mathcal {H}\)  (or equivalently \(\mathcal {F}\) ). For the computation of the Karcher mean, we require tools from differential geometry called the exponential and inverse-exponential maps. The exponential map at a point \(h_1 \in \mathcal {H}\) , denoted by \(\exp : T_{h_1}(\mathcal {H}) \mapsto \mathcal {H}\) , is defined as \( \exp _{h_1}(\delta h) = \cos (\Vert  \delta h \Vert )h_1+ \sin (\Vert  \delta h \Vert )(\delta h/\Vert \delta h\Vert )\) ,
where \(\Vert \delta h\Vert =\big (\int _0^1 \delta h(x)^2 dx\big )^{1/2}\) . The inverse-exponential map, denoted by \(\exp ^{-1}_{h_1}: \mathcal {H} \mapsto T_{h_1}(\mathcal {H})\) , is given by \(\exp ^{-1}_{h_1}(h_2) = (\theta /\sin (\theta ))(h_2 - h_1\cos (\theta ))\) .
